Factoring Practice: Greatest Common Factor (GCF)

Part 1: The Step-by-Step Guide

First, what are Factors?

Think of factors as numbers you can multiply together to get another number.

Example:

The factors of 12 are 1, 2, 3, 4, 6, 12 because:

1 × 12 = 12
2 × 6 = 12
3 × 4 = 12

What is the Greatest Common Factor (GCF)?

The GCF is the biggest factor that two or more numbers share.

Example: Find the GCF of 12 and 18.

Factors of 12: 1, 2, 3, 4, 6, 12

Factors of 18: 1, 2, 3, 6, 9, 18

The factors they have in common are 1, 2, 3, and 6. The biggest (greatest) one is 6. So, the GCF is 6!

How to Factor the GCF Out of a Polynomial

This is like doing the distributive property backwards. We pull out the GCF from all the terms.

Let's Factor: 8x + 12

Step 1: Find the GCF of the numbers.

Factors of 8: 1, 2, 4, 8
Factors of 12: 1, 2, 3, 4, 6, 12
The GCF of the numbers is 4.

Step 2: Check for common variables.

The first term has an 'x' (8x).
The second term does not have an 'x' (12).
Since they don't both have an 'x', we can't pull out a variable.
So, the total GCF is just 4.

Step 3: Write the GCF outside parentheses.

4 ( )

Step 4: Divide each original term by the GCF to see what's left.

First term: 8x ÷ 4 = 2x
Second term: 12 ÷ 4 = 3

Step 5: Put the results from Step 4 inside the parentheses.

4 (2x + 3)

So, the final factored answer is 4(2x + 3).

You can check your answer by distributing: 4 * 2x = 8x and 4 * 3 = 12. You get back to `8x + 12`!

Example with Variables

When terms share variables, find the smallest exponent for that variable. That's the GCF for the variable part.

Let's Factor: 15x³ + 25x²

Step 1: GCF of the numbers (15 and 25).

The GCF is 5.

Step 2: GCF of the variables (x³ and x²).

They both have an 'x'.
The exponents are 3 and 2. The smallest one is 2.
So, the variable GCF is x².

Combine them: The total GCF is 5x².

Step 3 & 4: Divide each term by the GCF (5x²).

First term: 15x³ ÷ 5x² = 3x (Remember: 15÷5=3 and x³÷x²=x)
Second term: 25x² ÷ 5x² = 5

Step 5: Write the final answer.

5x² (3x + 5)

Part 2: Practice Problems

Now it's your turn! Take your time and follow the steps. You can do it!

1. Find the GCF of the numbers:

a) 10 and 25

b) 18 and 24

c) 16, 24, and 40

2. Find the GCF of the terms (numbers and variables):

a) 4x and 8

b) 6x² and 9x

c) 7y³ and 14y⁵

3. Factor the GCF out of the polynomial:

a) 3x + 12

4. Factor the GCF out of the polynomial:

b) 10y - 15

5. Factor the GCF out of the polynomial:

c) 6x + 24y

6. Factor the GCF out of the polynomial (with exponents):

d) 8a² + 4a

7. Factor the GCF out of the polynomial (with exponents):

e) 5n³ - 20n

8. Challenge! Factor the GCF out of this three-term polynomial:

f) 9x³ + 6x² + 12x